On the Identity of Some von Neumann Algebras and Some Consequences

Among von Neumann algebras, the Weyl algebra W{\mathcal{W}} generated by two unitary groups {U(α)} and {V(β)}, the algebra U{\mathcal{U}} generated by a completely non-unitary semigroup of isometries {U ₊(α)} and the Weyl algebra W₊^h{\mathcal{W}_{+}^{h}} pertaining to a semi-bounded space with homogeneous spectrum of the generator of {V(β)}, all share the property that their representations are completely reducible and the irreducible representations are equivalent. We trace this fact to the identity of these algebras, in the sense that any of them contains a representation of any of the remaining two algebras, which in turn contains the original algebra. We prove this statement by explicit construction. The aforementioned results about the representations of the algebras follow immediately from the proof for any of them. Also, by the above construction we prove for W^h₊{\mathcal{W}^{h}_{+}} the analog of a theorem by Sinai for W{\mathcal{W}} : given {V(β)} with semi-bounded homogeneous spectrum, there exists a completely non-unitary semigroup {U ₊(α)} such that {V(β)} and {U ₊(α)} generate W₊^h{\mathcal{W}_{+}^{h}}.     

Equivalence of deformations and associated *-products

In this letter we study the non-trivial formal differentiable deformations of the Lie algebraN=C (W, IR) whereW is a symplectic manifold. Under some assumptions (satisfied in particular forW=IR²ⁿ ) we show that these deformations are all equivalent, up to a monomial change of the parameter, to one of them (Moyal for IR²ⁿ ). Furthermore, if there exists a differentiable *-product corresponding to one of them, each of them is induced by a *-product which is essentially unique.Aspirant du Fonds National belge de la Recherche Scientifique.     

Quantization of Kähler manifolds. III

We use Berezin's dequantization procedure to define a formal *-product on the algebra of smooth functions on the unit disk in ℂ. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.     

Quantization and representation theory of finiteW algebras

In this paper we study the finitely generated algebras underlyingW algebras. These so called finiteW algebras are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings ofsl ₂ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finiteW algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finiteW symmetry. In the second part we BRST quantize the finiteW algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finiteW algebras in one stroke. Examples are given. In the last part of the paper we study the representation theory of finiteW algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finiteW algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finiteW algebras.     

The relation between quantumW algebras and Lie algebras

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrarysl ₂ embeddings we show that a large set of quantumW algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set contains many knownW algebras such asW _N andW ₃ ⁽²⁾ . Our formalism yields a completely algorithmic method for calculating theW algebra generators and their operator product expansions, replacing the cumbersome construction ofW algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that anyW algebra in can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Thereforeany realization of this semisimple affine Lie algebra leads to a realization of theW algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolusions for all algebras in. Some examples are explicitly worked out.     

Star-products and phase space realizations of quantum groups

It is shown for a family of *-products (i.e. different ordering rules) that, under a strong invariance condition, the functions of the quadratic preferred observables (which generate the Cartan subalgebra in phase space realization of Lie algebras) take only the linear or exponential form. An exception occurs for the case of a symmetric ordering *-product where trigonometric functions and two special polynomials can also be included. As an example, the quantized algebra of the oscillator Lie algebra is argued.     

W-algebras and superalgebras from constrained WZW models: A group theoretical classification

We present a classification ofW algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with anSl(2) subalgebra (resp.OSp(1/2) superalgebra) of a simple Lie algebra (resp. superalgebra)G. However, the determination of anU(1) _Y factor, commuting withSl(2) (resp.OSp(1/2)), appears, when it exists, particularly useful to characterize the correspondingW algebra. The (super) conformal spin contents of eachW (super) algebra is performed. The class of all the superconformal algebras (i.e. with conformal spinss<=2) is easily obtained as a byproduct of our general results.     

An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a ^*-product on the cotangent bundle T ^* G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a ^*-product on IR²ⁿ.Chargé de recherches au FNRS, on leave of absence from Université libre de Bruxelles.     

Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Transitive Novikov Algebras on Four-Dimensional Nilpotent Lie Algebras

Novikov algebras were introduced in connection with the Poisson brackets (of hydrodynamic type) and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra, and the radical of a finite-dimensional Novikov algebra is transitive. In this paper, we give a classification of transitive Novikov algebras on four-dimensional nilpotent Lie algebras based on Kim (1986, Journal of Differential Geometry 24, 373–394).     

Vertex Operators Arising from the Homogeneous Realization for

We use the underlying Fock space for the homogeneous vertex operator representation of the affine Lie algebra to construct a family of vertex operators. As an application, an irreducible module for an extended affine Lie algebra of type A _{N −1} coordinatized by a quantum torus ℂ _q of 2 variables (or 3 variables) is obtained. Moreover, this module turns out to be a highest weight module which is an analog of the basic module for affine Lie algebras. Received: 16 August 1999 / Accepted: 18 January 2000     

Real forms of extended Kac–Moody symmetries and higher spin gauge theories

We consider the relation between higher spin gauge fields and real Kac–Moody Lie algebras. These algebras are obtained by double and triple extensions of real forms \mathfrakg₀{\mathfrak{g}_0} of the finite-dimensional simple algebras \mathfrakg{\mathfrak{g}} arising in dimensional reductions of gravity and supergravity theories. Besides providing an exhaustive list of all such algebras, together with their associated involutions and restricted root diagrams, we are able to prove general properties of their spectrum of generators with respect to a decomposition of the triple extension of \mathfrakg₀{\mathfrak{g}_0} under its gravity subalgebra \mathfrakgl(D,\mathbb R){\mathfrak{gl}(D,\mathbb {R})} . These results are then combined with known consistent models of higher spin gauge theory to prove that all but finitely many generators correspond to non-propagating fields and there are no higher spin fields contained in the Kac–Moody algebra.     

Tensor operators for quantum algebras

Tensor operators are discussed for Hopf algebras and, in particular, for a quantum (q-deformed) algebraUq(g), whereg is any simple finite-dimensional or affine Lie algebra. These operators are defined via an adjoint action in a Hopf algebra. There are two types of the tensor operators which correspond to two coproducts in the Hopf algebra. In the case of tensor products of two tensor operators one can obtain 8 types of the tensor operators and so on. We prove the relations which can be a basis for a proof of the Wigner-Eckart theorem for the Hopf algebras. It is also shown that in the case ofUq(g) a scalar operator can be differed from an invariant operator but atq=1 these operators coincide. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Supported by Russian Foundation for Fundamental Research, grant 99-01-01163, and by INTAS-00-00055.     

Universal Construction of Algebras

We present a direct construction of the abstract generators for q-deformed W_N{\cal W}_N algebras. New quantum algebraic structures of W_q,p{\cal W}_{q,p} type are thus obtained. This procedure hinges upon a twisted trace formula for the elliptic algebra \elp\elp generalizing the previously known formulae for quantum groups. It represents the q-deformation of the construction of W_N{\cal W}_N algebras from Lie algebras.     

{\hbar}-adic Quantum Vertex Algebras and Their Modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of (^h/_2p){\hbar}-adic nonlocal vertex algebra and (^h/_2p){\hbar}-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free \mathbb C[[(^h/_2p)]]{{\mathbb C}\lbrack\lbrack{\hbar}\rbrack\rbrack}-module W, we study (^h/_2p){\hbar}-adically compatible subsets and (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subsets of (End W)[[x, x ⁻¹]]. We prove that any (^h/_2p){\hbar}-adically compatible subset generates an (^h/_2p){\hbar}-adic nonlocal vertex algebra with W as a module and that any (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subset generates an (^h/_2p){\hbar}-adic weak quantum vertex algebra with W as a module. A general construction theorem of (^h/_2p){\hbar}-adic nonlocal vertex algebras and (^h/_2p){\hbar}-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of \mathfrak s\mathfrak l₂{{\mathfrak s}{\mathfrak l}_{2}} to (^h/_2p){\hbar}-adic quantum vertex algebras.     

Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is obtained from the unrenormalized one by evaluating at ɛ= 0 the holomorphic part γ₊(ɛ) of the Riemann–Hilbert decomposition γ₋(ɛ)^{− 1}γ₊(ɛ) of the loop γ(ɛ)∈G provided by dimensional regularization. We show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. More precisely, the formula g ₀=gZ ₁ Z ₃ ^−3/2 for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra ℋ. This allows first of all to read off directly, without using the group G, the bare coupling constant and the renormalized one from the Riemann–Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter ɛ. It also allows to lift both the renormalization group and the β-function as the asymptotic scaling in the group G. This exploits the full power of the Riemann–Hilbert decomposition together with the invariance of γ₋(ɛ) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue. Received: 21 March 2000 / Accepted: 3 October 2000     

The algebra ,η( ) and infinite Hopf family of algebras

New deformed affine algebras, _,η( ), are defined for any simply laced classical Lie algebra g, which are generalizations of the algebra, _,η( ₂), recently proposed by Khoroshkin-Lebedev-Pakuliak (KLP). Unlike the work of KLP, we associate with the new algebras the structure of an infinite Hopf family of algebras in contrast to the one containing only finite number of algebras, introduced by KLP. Bosonic representation for _,η( ) at level 1 is obtained, and it is shown that, by repeated application of Drinfeld-like comultiplications, a realization of _,η( ) at any positive integer level can be obtained. For the special case of g = sl_r+1, (r + 1)-dimensional evaluation representation is given. The corresponding interwining operations are also discussed.     

Dynamical symmetries for superintegrable quantum systems

We study the dynamical symmetries of a class of two-dimensional superintegrable systems on a 2-sphere, obtained by a procedure based on the Marsden-Weinstein reduction, by considering its shape-invariant intertwining operators. These are obtained by generalizing the techniques of factorization of one-dimensional systems. We firstly obtain a pair of noncommuting Lie algebras su(2) that originate the algebra so(4). By considering three spherical coordinate systems, we get the algebra u(3) that can be enlarged by “reflexions” to so(6). The bounded eigenstates of the Hamiltonian hierarchies are associated to the irreducible unitary representations of these dynamical algebras. The text was submitted by the authors in English.     

Quantization of K?hler manifolds. IV

We use Berezin's dequantization procedure to define a formal *-product on the algebra of smooth functions on the bounded symmetric domains. We prove that this formal *-product is convergent on a dense subalgebra of the algebra of smooth functions.This work was partially supported by EC contract CHRX-CT92-0050.